Monday, October 19, 2015

Reading List: The Road to Relativity

One hundred years ago, in 1915, Albert Einstein published the final
version of his general theory of relativity, which extended his 1905
special theory to encompass accelerated motion and gravitation. It
replaced the Newtonian concept of a “gravitational force”
acting instantaneously at a distance through an unspecified mechanism
with the most elegant of concepts: particles not under the influence
of an external force move along spacetime
geodesics, the
generalisation of straight lines, but the presence of mass-energy
curves spacetime, which causes those geodesics to depart from straight
lines when observed at a large scale.

For example, in Newton's conception of gravity, the Earth orbits the Sun
because the Sun exerts a gravitational force upon the Earth which pulls it
inward and causes its motion to depart from a straight line. (The Earth also
exerts a gravitational force upon the Sun, but because the Sun is so much
more massive, this can be neglected to a first approximation.) In general
relativity there is no gravitational force. The Earth is moving in a straight
line in spacetime, but because the Sun curves spacetime in its vicinity this
geodesic traces out a helix in spacetime which we perceive as the Earth's
orbit.

Now, if this were a purely qualitative description, one could dismiss it
as philosophical babble, but Einstein's theory provided a precise description
of the gravitational field and the motion of objects within it and, when
the field strength is strong or objects are moving very rapidly, makes
different predictions than Newton's theory. In particular, Einstein's theory
predicted that the perihelion of the orbit of Mercury would rotate around the
Sun more rapidly than Newton's theory could account for, that light propagating
near the limb of the Sun or other massive bodies would be bent through twice the
angle Newton's theory predicted, and that light from the Sun or other
massive stars would be red-shifted when observed from a distance. In due
course all of these tests have been found to agree with the predictions of
general relativity. The theory has since been put to many more precise
tests and no discrepancy with experiment has been found.
For a theory which is, once you get past the cumbersome
mathematical notation in which it is expressed, simple and elegant, its
implications are profound and still being explored a century later.
Black holes,
gravitational lensing,
cosmology and the large-scale
structure of the universe,
gravitomagnetism,
and gravitational radiation
are all implicit in Einstein's equations, and exploring them are among
the frontiers of science a century hence.

Unlike Einstein's original 1905
paper on special
relativity, the 1915 paper, titled
“Die Grundlage der allgemeinen
Relativitätstheorie” (“The Foundation of General
Relativity”) is famously difficult to comprehend and baffled many
contemporary physicists when it was published. Almost half is a tutorial
for physicists in
Riemann's
generalised
multidimensional geometry and the
tensor language
in which it is expressed. The balance of the paper is written in this
notation, which can be forbidding until one becomes comfortable with
it.

That said, general relativity can be understood intuitively the same way
Einstein began to think about it: through thought experiments. First,
imagine a person in a stationary elevator in the Earth's gravitational
field. If the elevator cable were cut, while the elevator was in free
fall (and before the sudden stop), no experiment done within the elevator
could distinguish between the state of free fall within Earth's gravity
and being in deep space free of gravitational fields. (Conversely, no
experiment done in a sufficiently small closed laboratory can distinguish
it being in Earth's gravitational field from being in deep space accelerating
under the influence of a rocket with the same acceleration as Earth's gravity.)
(The “sufficiently small” qualifier is to eliminate the effects
of tides, which we can neglect at this level.)

The second thought experiment is a bit more subtle. Imagine an observer
at the centre of a stationary circular disc. If the observer uses rigid
rods to measure the radius and circumference of the disc, he will find
the circumference divided by the radius to be 2π, as expected from
the Euclidean geometry of a plane. Now set the disc rotating and repeat
the experiment. When the observer measures the radius, it will be as
before, but at the circumference the measuring rod will be contracted
due to its motion according to special relativity, and the circumference,
measured by the rigid rod, will be seen to be larger. Now, when the circumference
is divided by the radius, a ratio greater than 2π will be found, indicating
that the space being measured is no longer Euclidean: it is curved. But
the only difference between a stationary disc and one which is rotating is
that the latter is in acceleration, and from the reasoning of the first
thought experiment there is no difference between acceleration and gravity.
Hence, gravity must bend spacetime and affect the paths of objects (geodesics)
within it.

Now, it's one thing to have these kinds of insights, and quite another to
puzzle out the details and make all of the mathematics work, and this
process occupied Einstein for the decade between 1905 and 1915, with many
blind alleys. He eventually came to understand that it was necessary to
entirely discard the notion of any fixed space and time, and express the
equations of physics in a way which was completely independent of any
co-ordinate system. Only this permitted the metric structure of
spacetime to be completely determined by the mass and energy within it.

This book contains a facsimile reproduction of Einstein's original
manuscript, now in the collection of the Hebrew University of Jerusalem.
The manuscript is in Einstein's handwriting which, if you read German,
you'll have no difficulty reading. Einstein made many edits to the
manuscript before submitting it for publication, and you can see them all
here. Some of the hand-drawn figures in the manuscript have been cut
out by the publisher to be sent to an illustrator for preparation of
figures for the journal publication. Parallel to the manuscript, the
editors describe the content and the historical evolution of the concepts
discussed therein. There is a 36 page introduction which describes the
background of the theory and Einstein's quest to discover it and the
history of the manuscript. An afterword provides an overview of
general relativity after Einstein and brief biographies of principal
figures involved in the development and elaboration of the theory.
The book concludes with a complete English translation of Einstein's
two papers given in the manuscript.

This is not the book to read if you're interested in learning general
relativity; over the last century there have been great advances in
mathematical notation and pedagogy, and a modern text is the best
resource. But, in this centennial year, this book allows you to
go back to the source and understand the theory as Einstein presented it,
after struggling for so many years to comprehend it. The supplemental
material explains the structure of the paper, the essentials of the
theory, and how Einstein came to develop it.